Abstract:
A \emph{Mathematica} Notebook is presented which allows for the transfer or any kind of polynomial expression to \emph{Matlab}. The output is formatted in such a way that \emph{Matlab} routines such as "Root" can be readily implemented. Once the Notebook has been executed, only one copy-paste operation in necessary.

Abstract:
We present a \emph{Mathematica} notebook allowing for the symbolic calculation of the $3\times3$ dielectric tensor of a electron-beam plasma system in the fluid approximation. Calculation is detailed for a cold relativistic electron beam entering a cold magnetized plasma, and for arbitrarily oriented wave vectors. We show how one can elaborate on this example to account for temperatures, arbitrarily oriented magnetic field or a different kind of plasma.

Abstract:
Proton beam transport in the context of proton driven Fast Ignition is usually assumed to be stable due to protons high inertia, but an analytical analysis of the process is still lacking. The stability of a charge and current neutralized proton beam passing through a plasma is therefore conducted here, for typical proton driven Fast Ignition parameters. In the cold regime, two fast growing Buneman-like modes are found, with an inverse growth-rate much smaller than the beam time-of-flight to the target core. The stability issue is thus not so obvious, and Kinetic effects are investigated. One unstable mode is found stabilized by the background plasma protons and electrons temperatures. The second mode is also damped, providing the proton beam thermal spread is larger than $\sim$ 10 keV. In Fusion conditions, the beam propagation should therefore be stable.

Abstract:
The filamentation instability occurring when a non relativistic electron beam passes through a quantum magnetized plasma is investigated by means of a cold quantum magnetohydrodynamic model. It is proved that the instability can be completely suppressed by quantum effects if and only if a finite magnetic field is present. A dimensionless parameter is identified which measures the strength of quantum effects. Strong quantum effects allow for a much smaller magnetic field to suppress the instability than in the classical regime.

Abstract:
Unstable modes growing when two plasma shells cross over a background plasma at arbitrary angle $\theta$, are investigated using a non-relativistic three cold fluids model. Parallel flows with $\theta=0$ are slightly more unstable than anti-parallel ones with $\theta=\pi$. The case $\theta=\pi/2$ is as unstable as the $\theta=0$ one, but the fastest growing modes are oblique. While the most unstable wave vector varies with orientation, its growth rate slightly evolves and there is no such thing as a stable configuration. A number of exact results can be derived, especially for the $\theta=\pi/2$ case.

Abstract:
We present an investigation of the electromagnetic instabilities which are trig gered when an ultra relativistic electron beam passes through a plasma. The linear growth rate is computed for every direction of propagation of the unstable modes, and temperatures are modelled using simple waterbag distribution functions. The ultra relativistic unstable spectrum is located around a very narrow band centered on a critical angle which value is given analytically. The growth rate of modes propagating in this direction decreases like k^(-1/3).

Abstract:
Many competing linear instabilities are likely to occur in astrophysical settings, and it is important to assess which one grows faster for a given situation. An analytical model including the main beam plasma instabilities is developed. The full 3D dielectric tensor is thus explained for a cold relativistic electron beam passing through a cold plasma, accounting for a guiding magnetic field, a return electronic current and moving protons. Considering any orientations of the wave vector allows to retrieve the most unstable mode for any parameters set. An unified description of the Filamentation (Weibel), Two-Stream, Buneman, Bell instabilities (and more) is thus provided, allowing for the exact determination of their hierarchy in terms of the system parameters. For relevance to both real situations and PIC simulations, the electron-to-proton mass ratio is treated as a parameter, and numerical calculations are conducted with two different values, namely 1/1836 and 1/100. In the system parameters phase space, the shape of the domains governed by each kind of instability is far from being trivial. For low density beams, the ultra-magnetized regime tends to be governed by either the Two-Stream or the Buneman instabilities. For beam densities equalling the plasma one, up to four kinds of modes are likely to play a role, depending of the beam Lorentz factor. In some regions of the system parameters phase space, the dominant mode may vary with the electron-to-proton mass ratio. Application is made to Solar Flares, Intergalactic Streams and Relativistic shocks physics.

Abstract:
The concept of energy balance is a key one in climate science. Yet, students may find it counterintuitive: while it is obvious that some energy comes in from the sun, the part coming out is more elusive. Asking them why the earth is not burning after billions of years of exposure to the sun, takes them to the question "where does the energy goes?" A series of Fermi like calculations then convinces them that storage capabilities are negligible compared to the amount of energy coming in: the earth necessarily re-emits what it receives.

Abstract:
The growth rate of the filamentation instability triggered when a diluted cold electron beam passes through a cold plasma is evaluated using the quantum hydrodynamic equations. Compared with a cold fluid model, quantum effects reduce both the unstable wave vector domain and the maximum growth rate. Stabilization of large wave vector modes is always achieved, but significant reduction of the maximum growth rate depends on a dimensionless parameter that is provided. Although calculations are extended to the relativistic regime, they are mostly relevant to the non-relativistic one.

Abstract:
The Rayleigh-Taylor instability is a key process in many fields of Physics ranging from astrophysics to inertial confinement fusion. It is usually analyzed deriving the linearized fluid equations, but the physics behind the instability is not always clear. Recent works on this instability allow for an very intuitive understanding of the phenomenon and for a straightforward calculation of the linear growth rate. In this Letter, it is shown that the same reasoning allows for a direct derivation of the relativistic expression of the linear growth rate for an incompressible fluid.